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Let [itex]f:[0,\infty)\rightarrow\mathbb{R}[/itex] be a bounded measurable function such that

[tex]\lim_{x\rightarrow\infty}x^2f(x)=1.[/tex]

Find an integral expression for

[tex]\lim_{\lambda\rightarrow 0^+}\frac{\int_0^{\infty}(1-\cos(x))f(\frac{x}{\lambda})dx}{\lambda^2}.[/tex]

This one is really bizarre to me. I'm not sure how to use the information about f's endpoint behavior other than to try somehow to approximate f by 1/x^2?

I've tried changing lambda into 1/n where n goes to infinity but that seems to get me an expression that will tend to 0, not an integral expression as requested.

Any help or ideas would be greatly appreciated; a solution isn't necessary, but maybe thoughts on how to think about the problem?

Thanks!

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# Prelim problem: bizarre integral expression

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